The 57-year-old is taking over from fellow Australian Dan McKellar, who parted ways with the club last week.
Cheika was previously head coach of Argentina for two years before stepping down in December. He led the team to fourth place at last year's Rugby World Cup.
"Honestly, I wasn't looking at the Premiership and didn't have the desire to coach in it until Leicester Tigers came to me," Cheika said in a statement.
"But, the opportunity to coach at Tigers and lead this group of players turned my head."
Cheika won the European Cup with Leinster, the Super Rugby title with NSW Waratahs and also guided Australia to the Rugby World Cup final in 2015 and to the quarter-finals four years later.
"We believe Michael is the right person to take this team, this club back to where we know we should be," Leicester CEO Andrea Pinchen said in a statement.
"I also accept that the past week is not what fans expected, and nor did we, but the decisions we have made are for the long-term benefit of Leicester Tigers.
"And after lengthy, detailed and very honest conversations with Michael in recent days, we are on the same page about what is now necessary to see this club back on top."
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